Drill strings are complex dynamical systems with many uncertain parameters. The drill string interaction with the borehole produces a variety of undesired oscillations. The stickslip phenomenon is the extreme state of torsional vibrations, which causes the drill string to stop rotating and then spin free periodically. This non-uniform rotation may cause the wear of expensive equipment or even catastrophic failures in drill strings. Therefore, it is essential to study the drilling parameters in order to develop appropriate control approach for the suppression of the stick-slip vibration. However, the complexity of the drill string system poses several modeling and control challenges. The drill string model challenges include thermal, physical, electrical, and environmental influences on the stick-slip, simple enough to perform the analysis and control purposes. The control challenges include dealing with the complex dynamics of nonlinear friction, minimize nonlinear torque on the bit, and perform more robust during operating conditions. The control techniques are divided into two major approaches: passive and active control approaches. The passive control approaches include design sophisticated bits (with depth of cut control technology) to limit the reactive torque that might lead to the stick-slip, optimizing the drilling parameters, and using antivibration down hole tools. The active control approaches are on active anti-vibration control methods due to the improvements in the real-time measurement and control systems. Two of the most common active control techniques used in drill string system are proportional-derivative and sliding mode control methods. This paper presents an overview and a comparative study of the common control methods belonging to the common active control methods to mitigate the stick-slip phenomenon in drill string systems. The main objective is to assess the impact of the active control approaches to mitigate the stick-slip phenomenon. First, the common model for drillstring system is presented. Then, the study presents analyses of different drilling parameters, such as the weight on bit (WOB) and associated torque on bit (TOB) that define the bit aggressiveness, which are key in mitigating stick-slip vibration. These parameters have been considered as the comparison factors. Furthermore, this study details the design process of these controllers, and evaluates the performances of the different control systems to track the reference signal of bit velocity taking into account parametric uncertainties. Discussion and recommendation about the drilling parameters optimization are presented. This paper provides the necessary information needed for modeling and control of drillstring systems with minimum stick-slip vibrations. The results show that the adaptive sliding mode controller succeeded to eliminate the stick-slip phenomenon with better robustness to parametric uncertainties and weight on bit variations compared to the other controllers.